A Note on the Convolution of the Uniform and Related ... - CiteSeerX

c Heldermann Verlag
ISSN 0940-5151

Economic Quality Control Vol 16 (2001), No. 1, 17 – 41

A Note on the Convolution of the Uniform and Related Distributions and Their Use in Quality Control Frank Killmann and Elart von Collani

Abstract: Consider a product with quality characteristic X. Assume that the product is composed of n parts each with quality characteristic Xi , and and let X = X1 + · · · + Xn Assume that there is a lower specification L and an upper specification U for X. Then the problem arises how to select suitable specifications for the quality characteristics Xi of the parts. If the type of distributions of the partial quality characteristics Xi are known, then for solving the specification problem, the distribution of X = X1 + · · · + Xn is needed. In case of measurable quality features, usually the normal model is assumed for the Xi and the problem of deriving the distribution of X is solved in the well-known way. However, generally the only feature of Xi which is known with certainty is its bounded support, which follows from technical conditions. In such a case the normal approximation which is based on an unbounded support of each Xi may lead to a distribution of X which does not reflect reality sufficiently well. In this paper, the uniform or a related distribution is assumed for the partial quality characteristics and an explicit expression for the distribution of the sum X is derived. The usefulness of the result is illustrated by an example taken from industrial practice. Key Words: Uniform Distribution, Convolution, Statistical Tolerancing, Production-oriented Tolerancing

1

Introduction

Let X be a random variable which may be represented by a sum of n random variables X1 , · · · , Xn : X = X1 + · · · + Xn Such an additive relation of random variables is often found in real world situations. Simple examples are:

18

Frank Killmann and Elart von Collani

• Length or weight of a chain with n links. • Electric resistance of a series circuit. • Transmission time for an information split up into packages. • The total life length of n products, where the next one is put on operation after failure of the preceding one. In this paper it is assumed that the summands Xi , i = 1, · · · , n are independent and uniformly distributed random variables. 1.1 Definition A random variable Y is said to be uniformly distributed over the interval [a; b], denoted Y ∼ U [a; b], if its density function is given by:
1 for a ≤ y ≤ b b−a (1) fY (y) = 0 otherwise or equivalently, its distribution function by:  for y < a  0 y−a for a ≤ y < b FY (y) =  b−a 1 for b ≤ y.

(2) •

fY (y)6

FY (y)6 1

1 b−a

-

a b y Fig. 1: Density function

-

a b y Fig. 2: Distribution function

The interval [a; b] is called support of Y and the expectation and the variance of Y are given by (b − a)2 a+b and V [Y ] = (3) 2 12 For a detailed overview about the uniform distribution and its properties, we refer to [6]. E[Y ] =

The uniform distribution can be characterized by means of the principle of maximum entropy.

A Note on the Convolution of the Uniform and Related Distributions

19

1.2 Definition The entropy H of a distribution with density function f (x) is given by  +∞ f (t) log f (t) dt (4) H(f ) := − −∞

•

Then the uniform distribution has the following property: 1.3 Theorem The uniform distribution has maximum entropy among all distributions of continuous type with finite support [a; b]. • Hence, in case that a finite range [a; b] is the only information about the random variable of interest, the use of any distribution different from the uniform one would be either tantamount to assume further unconfirmed information or to neglect confirmed information. Using e.g. the normal distribution in case of finite support means to neglect given information. The information ’finite support’ is reduced to ’finite variance’, in which case the principle of maximum entropy yields the normal distribution: Finite support Finite variance

2

MEP

−→

Uniform Distribution

−→

Normal Distribution

MEP

Convolution of Uniform Distributions

Consider a sum X of independent and uniformly distributed random variables Xi ∼ U [ai , bi ], i = 1, . . . , n: X = X1 + · · · + Xn

(5)

then the following is true. • The sum X is symmetrically distributed around

(a1 +...+an )+(b1 +...bn ) . 2

• If n 

(bi − ai )2 −→ ∞ n→∞

(6)

i=1

then by the central limit theorem the distribution of n  X− E[Xi ] i=1 n  V [Xi ] i=1

tends for n → ∞ to the standard normal distribution.

(7)

20

Frank Killmann and Elart von Collani

Hence in case (6) and for sufficiently large n the distribution of X can be approximated 

n n  by the normal distribution N E[Xi ], V [Xi ] . However, it turns out that in spite i=1

i=1

of the symmetry relation convergence to the normal distribution may be rather slow, if the lengths bi − ai are very different implying that the normal approximation might be bad even for large values of n. Therefore, the exact distribution of X would be desirable. 2.1

Convolution of Identical Uniform Distributions

Historically, the case of independent and identically uniformly distributed Xi s played an important role, i. e. X = X1 + · · · + Xn

with Xi ∼ U [a; b], i = 1, . . . , n

(8)

According to R´enyi [9], p. 198, the distribution of X was first studied by N. I. Lobatchewski in 1842. ”He wanted to use it to evaluate the error of astronomical measurements, in order to decide whether the Euclidean or the non-Euclidean geometry is valid in the universe.” R´enyi [9], p. 197 also states the density function of X. 

n−1 n ˜ (n,x)   1 i n   (n−1)!(b−a) x − na − i(b − a) (−1) n i i=0 f (n) (x) =    0 where n ˜ (n, x) :=  x−na  = largest integer less than b−a

if na ≤ x ≤ nb (9) otherwise

x−na . b−a

It is well-known that the speed of convergence to the normal distribution is extremely fast in the identically distributed case given by (8). Already for n = 4 the difference between the normal approximation and the exact distribution is often negligible. If the single distributions are not identical, but have a common length of their support, i.e. bi − ai = b − a for i = 1, . . . , n, then it is possible to reduce the problem of deriving the distribution of X to the identically distributed case by suitable transformations. However, allowing arbitrary uniform distributions requires a different approach for determining the distribution function of X. 2.2

Convolution of Arbitrary Uniform Distributions

The convolution of arbitrary uniform distributions can be obtained by a result given in [1] which refers to the distribution function of a linear combination of independent U [0, 1]-distributed random variables. However, the given formula is rather unsuited for practical applications and, therefore, a different representation of the distribution function is derived here. In order to be able to give an explicit expression for distribution function of the sum of n independent and uniformly distributed random variables, some notations are needed.

A Note on the Convolution of the Uniform and Related Distributions

21

2.1 Notation For n ∈ IN0 let IBn denote the n-dimensional binary vector space {0, 1}n : n = 0 : IB0 = {
0}   n ≥ 1 : IBn =
j = (j1 , j2 , . . . , jn ) | ji ∈ {0, 1}, i = 1, 2, . . . , n On the space IBn a weight of a binary sequence is defined | . |: IBn → IN0

(10)

by
j →

n 

ji

(11)

i=1

Hence |
j | corresponds to the number of ones in the vector
j. Let ., . : IRn × IRn → IR

(12)

denote the Euclidean scalar product of IRn . Restricting the scalar product ., . on IBn ×IRn defines the function ., .
: IBn × IRn → IR

(13)

In the following the asterisk
is omitted and (13) is also denoted by ., . . For n = 0 we • have IB0 = IR0 = {
0} and 
0,
0 is defined to be 0. With theses notations, the distribution function FX (x) = FX1 +...+Xn (x) = F (n) (x) and the density function fX (x) = fX1 +...+Xn (x) = f (n) (x) are given below. 2.2 Theorem Let X1 , X2 , . . . , Xn be independent random variables with Xi ∼ U (ai , bi ) for i = 1, 2, . . . , n (n ∈ IN). Then

 0           (−1)|j| Kn (x − An − 
j,
b −
a )   j∈IBn−1 F (n) (x) =  n    n! (bi −ai )   i=1      1  0           (−1)|j| Kn−1 (x − An − 
j,
b −
a )   j∈IBn−1 f (n) (x) =  n    (n−1)! (bi −ai )   i=1      0

for x < An

for An ≤ x < Bn

(14)

for Bn ≤ x for x < An

for An ≤ x < Bn for Bn ≤ x

(15)

22

Frank Killmann and Elart von Collani

where An = a1 + a2 + · · · + an Bn = b1 + b2 + · · · + bn
(a2 , a3 , . . . , an ) for n > 1
an = (0) for n = 1

bn = (b2 , b3 , . . . , bn ) for n > 1 (0) for n = 1  for y < 0  0 n for 0 ≤ y < b1 − a1 y Kn (y) =  n y − (y − (b1 − a1 ))n for b1 − a1 ≤ y

(16) (17) (18) (19) (20) •

Proof: Obviously (0)

Xi ∼ U [ai , bi ] ⇔ Xi = Xi − ai ∼ U [0, bi − ai ] (21) n n   (0) Xi with distribution function F (n) (x) and X (0) = Xi with distribution Let X = i=1 0 F(n) (x).

i=1

(0)

function Moreover, An = a1 + . . . + an , Bn = b1 + . . . + bn , An n  (0) Bn = (bi − ai ).

= 0 and

i=1

Then



F (n) (x) = PX1 +...+Xn (−∞, x]

= PX (0) +...+Xn(0) (−∞, x − An ] 1

=

(n) F0 (x

− An )

(22)

Thus, it is sufficient to prove (14) for the special case that ai = 0 for i = 1, . . . , n. Therefore, consider the independent random variables Xi ∼ U [0, bi ] and their sum n  X= Xi with distribution function F (n) (x). Then, we have to prove that the following i=1

holds:

 0         1 (n) n  F (x) = n! bi  i=1        1

where

if x < 0  j∈IB

 (−1)|j| Kn (x − 
j,
bn )

if 0 ≤ x < Bn

n−1

if Bn ≤ x

(23)

23

A Note on the Convolution of the Uniform and Related Distributions


bn = (b2 , . . . , bn ) Bn = b1 +. . . + bn 0  yn Kn (y) =  n y − (y − b1 )n

for for for

y Bn n   |j|

(−1) Kn (x − j, bn ) = n! bi j∈IB

n−1

25

(31)

i=1

is true. Taking into account d Kn (x) = nKn−1 (x) dx or x Kn (x) − Kn (x) = n Kn−1 (y)dy

(32)

(33)

x

we obtain the following.   (−1)|j| Kn (x − 
j,
bn ) j∈IBn−1



=

Kn (x − j1 b2 − . . . − jn−1 bn )

j∈IBn−1

  (−1)|j| Kn (x − j1 b2 − . . . − jn−2 bn−1 − 0 · bn )−



=

jn−2 ∈IBn−2



=

 Kn (x − j1 b2 − . . . − jn−2 bn−1 − 1 · bn ) |j|

bn Kn−1 (x − 
j,
bn−1 − xn )dxn

(−1) n

j∈IBn−2

0

.. . bn bn−1 b2 = n! · · · K1 (x − x2 − . . . − xn )dx2 . . . dxn 0

0

(34)

0

For x > Bn = b1 + . . . + bn we have x − x2 − . . . − xn > b1

(35)

and, therefore, by the definition of Kn K1 (x − x2 − . . . − xn ) = b1

(36)

By inserting (36) into (34), we obtain for x > Bn : n   |j|

(−1) Kn (x − j, bn ) = n! bi j∈IB

n−1

i=1

which proves (31) and thus (30). Inserting (29) and (30) into (28) yields:

(37)

26

Frank Killmann and Elart von Collani

• x < 0: f (n+1) (x) =

1 bn+1

[0 − 0] = 0

(38)

• 0 ≤ x < Bn+1 : 

f (n+1) (x) =

   (−1)|j| Kn (x − 
j,
bn+1 ) − (−1)|j|−1 Kn (x − 
j,
bn+1 ) n n

j∈IB j∈IB jn =0

jn =1

n!

j∈IB

bi

i=1

 =

n+1 

n

 (−1)|j| Kn (x − 
j,
bn+1 )

n!

n+1 

(39) bi

i=1

• Bn+1 ≤ x: f (n+1) (x) =

1 [1 − 1] = 0 b+1

(40)

Hence

 0   
  (−1)|j| Kn (x−j,bn+1 )  n
j∈IB f (n+1) (x) = n+1   n! bi   i=1   1

for x < 0 for

0 ≤ x < Bn+1

(41)

for Bn+1 ≤ x

Integrating (41) we obtain by means of (32)  0     (−1)|
j| Kn+1 (x−j,bn+1 )   n
j∈IB (n+1) (x) = F n+1   (n+1)! bi   i=1   1

the distribution function F (n+1) (x). for x < 0 for

0 ≤ x ≤ Bn+1

(42)

for Bn+1 ≤ x •

which completes the proof. 2.3

The Case n = 2

Let X1 ∼ U [a1 , b1 ] and X2 ∼ U [a2 , b2 ], then three cases have to be distinguished. • Case I: a1 + b2 < a2 + b1 (2)

In this case Theorem 2.2 yields the following density function fI (x) of the sum X = X1 + X2 .

A Note on the Convolution of the Uniform and Related Distributions

(2)

fI (x) =

 0    x−(a1 +a2 )    (b1 −a1 )(b2 −a2 )      

1 b1 −a1 −x+(b1 +b2 ) (b1 −a1 )(b2 −a2 )

0

for for for for for

x < a1 + a2 a 1 + a 2 ≤ x < a1 + b 2 a1 + b2 ≤ x < a2 + b1 a 2 + b 1 ≤ x < b1 + b 2 b1 + b2 ≤ x

27

(43)

• Case II: a2 + b1 < a1 + b2 (2)

In Case II the following density function fII (x) of the sum X = X1 +X2 is obtained.  for x < a1 + a2   0 x−(a +a )  1 2   for a1 + a2 ≤ x < a2 + b1  (b1 −a1 )(b2 −a2 ) (2) 1 for a2 + b1 ≤ x < a1 + b2 (44) fII (x) = b2 −a2  −x+(b1 +b2 )   for a1 + b2 ≤ x < b1 + b2    (b1 −a1 )(b2 −a2 ) 0 for b1 + b2 ≤ x • Case III: a2 + b1 = a1 + b2 Finally, in the third case we obtain:  0 for x < a1 + a2     x−(a1 +a2 ) for a1 + a2 ≤ x < a2 + b1 = a1 + b2 (2) (b1 −a1 )(b2 −a2 ) fIII (x) = −x+(b1 +b2 )  for a1 + b2 = a2 + b1 ≤ x < b1 + b2    (b1 −a1 )(b2 −a2 ) 0 for b1 + b2 ≤ x

(45)

In the first two cases the density function has the shape of a trapezoid and in the third case the shape of a triangular. Since in Case I we have: (a1 + b2 ) − (a1 + a2 ) = b2 − a2 (b1 + b2 ) − (a2 + b1 ) = b2 − a2

(46) (47)

and analogously in Case II: (a2 + b1 ) − (a1 + a2 ) = b1 − a1 (b1 + b2 ) − (a1 + b2 ) = b1 − a1

(48) (49)

it follows that the trapezoid as well as the triangular are symmetric with respect to 1 +b2 ) x = (a1 +a2 )+(b . 2

3

Convolution of Related Distributions

The result with respect to uniformly distributed random variables may be utilized to obtain the distribution function of the sum of independent random variables each having a symmetric trapezoid or triangular distribution.

28

Frank Killmann and Elart von Collani

3.1 Definition A random variable Y is said to have a symmetric trapezoid distribution denoted by Y ∼ T ra[a; b; c], if its density function is given by:  0 for y < a        1 x−a  for a ≤ y < c  b−c c−a     1 for c ≤ y < b − (c − a) fY (y) = (50) b−c      1 −x+b  for b − (c − a) ≤ y < b   b−c c−a      0 b≤y • fY (y) 6

1 b−c

@

@

@

@

@ @

a

c

b − (c − a)

b

-

y

Fig. 3: Density function of a symmetric trapezoid distribution The symmetric triangular distribution is a special case of the symmetric trapezoid distribution. It is characterized by the relation c = b − (c − a) or, equivalently: a+b c= 2

(51)

(52)

3.2 Definition A random variable Y is said to have a symmetric triangular distribution denoted by Y ∼ T ri[a; b], if its density function is given by:  0 for y < a         1 x−a for a ≤ y < b+a  a+b b−a 2 (53) fY (y) =  a+b 1 −x+b  for ≤y